In the first sentence, we use 갑시다 which means "let's go" and 먹으러 갑시다 which means "let's go eat". However, I'd say 갑시다 is rather musculine in tone, rather confirmative and forceful and used usually by adults whereas 가요 is a polite form used by anyone who is talking to older people or anyone in any formal situations.

For example,

the boss of a company could say, 점심 먹으러 갑시다 to all his employee, "let's go eat lunch."